Key Concepts and Formulas

• Distributive Law: $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$ and $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
• Identity Laws: $p \vee \text{False} \equiv p$ and $p \wedge \text{True} \equiv p$
• Complement Laws: $p \vee \sim p \equiv \text{True}$ and $p \wedge \sim p \equiv \text{False}$
• Absorption Law: $p \vee (p \wedge q) \equiv p$ and $p \wedge (p \vee q) \equiv p$
• De Morgan's Laws: $\sim (p \vee q) \equiv \sim p \wedge \sim q$ and $\sim (p \wedge q) \equiv \sim p \vee \sim q$

Step-by-Step Solution

Step 1: Rewrite the given expression.

We are given the Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$. Our goal is to simplify this expression using Boolean algebra laws.

Step 2: Apply the distributive law.

We can rewrite the expression by grouping the last two terms: $(p \wedge \sim q) \vee (q \vee (\sim p \wedge q))$.

Step 3: Apply the absorption law to the second group.

Notice that $q \vee (\sim p \wedge q)$ can be simplified using the absorption law. Specifically, $q \vee (\sim p \wedge q) \equiv q$. This is because $\sim p \wedge q$ implies $q$, so adding $q$ to it doesn't change the value. Thus, we have $(p \wedge \sim q) \vee q$.

Step 4: Rearrange and apply the distributive law.

Rewrite the expression as $q \vee (p \wedge \sim q)$. We can rewrite this as $(q \vee p) \wedge (q \vee \sim q)$.

Step 5: Apply the complement law.

Since $q \vee \sim q \equiv \text{True}$, the expression becomes $(q \vee p) \wedge \text{True}$.

Step 6: Apply the identity law.

Using the identity law, $(q \vee p) \wedge \text{True} \equiv q \vee p$. Therefore, the expression simplifies to $p \vee q$.

Common Mistakes & Tips

• Carefully apply the distributive law. Make sure you are distributing correctly over $\vee$ and $\wedge$.
• Remember the absorption law: $p \vee (p \wedge q) = p$ and $p \wedge (p \vee q) = p$. Recognizing this can significantly simplify expressions.
• When in doubt, construct a truth table to verify your simplifications.

Summary

We started with the Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ and simplified it using Boolean algebra laws. We applied the absorption law, distributive law, complement law, and identity law to arrive at the simplified expression $p \vee q$.

Final Answer

The final answer is \boxed{p \vee q}, which corresponds to option (C).